Question: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-x-6y &= -1 \\ -2x-9y &= -6\end{align*}$
Answer: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-9y = 2x-6$ Divide both sides by $-9$ to isolate $y$ $y = {-\dfrac{2}{9}x + \dfrac{2}{3}}$ Substitute this expression for $y$ in the first equation. $-x-6({-\dfrac{2}{9}x + \dfrac{2}{3}}) = -1$ $-x + \dfrac{4}{3}x - 4 = -1$ Simplify by combining terms, then solve for $x$ $\dfrac{1}{3}x - 4 = -1$ $\dfrac{1}{3}x = 3$ $x = 9$ Substitute $9$ for $x$ back into the top equation. $- 9-6y = -1$ $-9-6y = -1$ $-6y = 8$ $y = -\dfrac{4}{3}$ The solution is $\enspace x = 9, \enspace y = -\dfrac{4}{3}$.